The main concern of this paper is to study large-time behavior of the sheath to the full Euler-Poisson system. As is well known, the monotone stationary solution under the Bohm criterion can be referred to as the sheath which is formed by interactions of plasma with wall. So far, the existence and asymptotic stability of stationary solutions in one-dimensional half space to the full Euler-Poisson system have been proved in \cite{DYZ2021}. In the present paper, we extend the results in \cite{DYZ2021} to $N$-dimensional ($N$=1,2,3) half space. By assuming that the velocity of the positive ion satisfies the Bohm criterion at the far field, we establish the global unique existence and the large time asymptotic stability of the sheath in some weighted Sobolev spaces by weighted energy method. Moreover, the time-decay rates are also obtained. A key different point from \cite{DYZ2021} is to derive some boundary estimates on the derivative of the potential in the $x_1$-direction.
The main purpose of this paper is to provide an effective procedure to study rigorously the relationship between unipolar and bipolar Euler-Poisson system in the perspective of mass. Based on the fact that the mass of an electron is far less than that of an ion, we amplify this property by letting $m_e/m_i\rightarrow 0$ and using two different singular limits to illustrate it, which are zero-electron mass limit and infinity-ion mass limit. We use the method of asymptotic expansion to handle the problem and find that the limiting process from bipolar to unipolar system is actually the process o…
This paper studies the two-dimensional Euler-Poisson equations associated with either attractive or repulsive forces. We mainly study the Riccati system that governs the flow's gradient. Under a suitable condition, it is shown that the Euler-Poisson system admits global smooth solutions for a large set of initial configurations. This paper is a continuation of our former work [8].
We prove shock formation for 3D Euler-Poisson system for electron fluid plasma. The shock solution we construct is of large initial data, also compactly supported during the lifespan. In addition, the blowup time and location can be computed explicitly.
We prove the unique existence of supersonic solutions of the Euler- Poisson system for potential flow in a three-dimensional rectangular cylinder when prescribing the velocity and the strength of electric field at the entrance. Overall, the main framework is similar to [1], but there are several technical differences to be taken care of vary carefully. And, it is our main goal to treat all the technical differences occurring when one considers a three dimensional supersonic solution of the steady Euler-Poisson system.
In this work, we study the behavior of blow-up solutions to the multidimensional restricted Euler--Poisson equations which are the localized version of the full Euler--Poisson system. We provide necessary conditions for the existence of finite-time blow-up solutions in terms of the initial data, and describe the asymptotic behavior of the solutions near blow up times. We also identify a rich set of the initial data which yields global bounded solutions.
We construct a special $C^{1,α}$ blow up solution to the three dimensional system modeling electro-hydrodynamics, which is strongly coupled with incompressible Euler equation and Nernst-Planck-Poisson equation. Our construction lies on the framework established in Elgindi [11] and relies on a special solution to variant spherical Laplacian.
Weak asymptotic methods for 3-D self-gravitating pressureless fluids. Application to the creation and evolution of solar systems from the fully nonlinear Euler-Poisson equations
This paper proves the existence of variational rotating solutions to the compressible non-isentropic Euler-Poisson equations with prescribed total mass. This extends the result of the isentropic case [Auchmuty and Beals, Arch. Ration. Mech. Anal., 1971] to the non-isentropic case. Compared with the previous result of variational rotating solutions in non-isentropic case [Wu, Journal of Differential Equations, 2015], to keep the constraint of a prescribed finite total mass, the author establishes a new variational structure the non-isentropic Euler-Poisson equations.
We show the continuity of the flow map for quasilinear symmetric hyperbolic systems with general right--hand sides in different functional setting, including weighted Sobolev spaces $H_{s,δ}$. An essential tool to achieve the continuity of the flow map is a low regularity energy estimate. We then apply these results to the Euler--Poisson system which describes various systems of physical interests.
In this note we derive the descriptions of the system of Euler-Poisson equations which governs the hydrodynamic evolution of gaseous stars in various co-ordinate systems. This note does not contain essentially new results for astrophysicists, but mathematically rigorous derivations cannot be found in the literatures written by physicists so that it will be useful to prepare details of rather stupidly honest derivations of the equations in various flames when we are going to push forward the mathematical research of the problem.
We consider several pressureless variants of the compressible Euler equation driven by nonlocal repulsionattraction and alignment forces with Poisson interaction. Under an energy admissibility criterion, we prove existence of global measure-valued solutions, i.e., very weak solutions described by a classical Young measure together with appropriate concentration defects. We then investigate the evolution of a relative energy functional to compare a measure-valued solution to a regular solution emanating from the same initial datum. This leads to a (partial) weak-strong uniqueness principle.
This paper deals with the existence of BV solution for the Euler-Poisson system endowed with a $γ$ pressure law. More precisely, we prove the existence of weak solution in the BV framework with arbitrary large initial data when $γ=1+2ε$ satisfies a smallness condition. We use the Glimm scheme combined with a splitting method as introduced in [Poupaud, Rascle and Vila, J. Differential Equations, 1995]. Existence of BV solution of 1-D isentropic Euler equation for large data and $γ=1+2ε$ is proved in [Nishida and Smoller, Comm. Pure Appl. Math, 1973]. Due to the presence of electric field, the d…
